Categorical bridge

In category theory, a discipline in mathematics, a bridge between categories $mathbb\; A$ and $mathbb\; B$ is a category $mathbb\; H$such that $mathbb\; A$ and $mathbb\; B$ are disjoint full subcategories of $mathbb\; H$ and $mathrm\{Ob\}mathbb\; H=mathrm\{Ob\}mathbb\; Acup\; mathrm\{Ob\}mathbb\; B$.Morphisms of $mathbb\; A$ and $mathbb\; B$ are called "homomorphisms" and the rest (passing "between" $mathbb\; A$ and $mathbb\; B$) are called heteromorphisms.

In notation: $mathbb\; H:mathbb\; Aleftrightharpoons\; mathbb\; B$.

As an example, the empty bridge between two categories is just their disjoint union.

A directed bridge from $mathbb\; A$ to $mathbb\; B$ is a bridge withoutarrows of the form $B\; o\; A$ (where $Binmathrm\{Ob\}mathbb\; B$ and $Ainmathrm\{Ob\}mathbb\; A$). We can easily see that directed bridges and profunctors(i.e. functors $F:mathbb\; A^\{op\}\; imesmathbb\; B\; omathrm\{Set\}$)are eventually the same [by identifying $F(A,B)$ with the set of heteromorphisms $A\; o\; B$] .

Bridge morphism

A morphism between bridges $mathbb\; H,\; mathbb\; K:mathbb\; Aleftrightharpoons\; mathbb\; B$is just a "functor" $varphi:mathbb\; H\; omathbb\; K$ which is identical on both$mathbb\; A$ and $mathbb\; B$, i.e. $varphimid\_\{mathbb\; A\}=\; mathrm\{id\}\_\{mathbb\; A\}$ and$varphimid\_\{mathbb\; B\}=\; mathrm\{id\}\_\{mathbb\; B\}$.

Profunctors (directed bridges)

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